Had a similar conversation with a game keeper about four years ago with much the same message although on rather more friendly terms.
Basically only a small section immediately behind the the cafe is Forestry commission land. Most of the rest of is owned by the duke of Norfolk estate with this then leased to local farms. Yes these farms and land are used for shoots in the season.
The game keeper basically told me we were welcome in small groups but to take care. Keep noise to a minimum to avoid scaring the nesting birds and wear bright clothing to avoid being shot at. Also be aware at night there will be marksmen present in the woods culling dear.
Basically they did not like noisy groups and aggressive trial building.